Efficient propagation of the hierarchical equations of motion using the Tucker and hierarchical Tucker tensors

Abstract

We develop new methods to efficiently propagate the hierarchical equations of motion (HEOM) by using the Tucker and hierarchical Tucker (HT) tensors to represent the reduced density operator and auxiliary density operators. We first show that by employing the split operator method, the specific structure of the HEOM allows a simple propagation scheme using the Tucker tensor. When the number of effective modes in the HEOM increases and the Tucker representation becomes intractable, the split operator method is extended to the binary tree structure of the HT representation. It is found that to update the binary tree nodes related to a specific effective mode, we only need to propagate a short matrix product state constructed from these nodes. Numerical results show that by further employing the mode combination technique commonly used in the multi-configuration time-dependent Hartree approaches, the binary tree representation can be applied to study excitation energy transfer dynamics in a fairly large system including over 104 effective modes. The new methods may thus provide a promising tool in simulating quantum dynamics in condensed phases.

Hierarchical equations of motion approach to hybrid fermionic and bosonic environments: Matrix product state formulation in twin space

“…A more suitable structure of the tensor train would reflect the natural structure of the super-Hamiltonian for the system of concern, which could be achieved by arranging highly entangled DoFs as close as possible and reducing the ranks to the smallest. 52,110 In addition, since the entanglement grows as time evolves, techniques that introduce rank adaptivity into the TDVP integration have been proposed, 102,111 which appear promising in practical applications.…”

“…Even with these advances, the method is limited to relatively small model systems, especially when the coupling to the environment is strong, mainly due to the factorial or exponential scaling with respect to the system and effective environmental DoFs. Recently, Shi and coworkers [50][51][52] as well as Borrelli and Gelin, 53,54 have separately suggested that the HEOM approach can be combined with the matrix product state (MPS) formulation, also called tensor train (TT) approach. The MPS formulation is an extremely powerful and versatile tool to study quantum many-body physics, in particular for one-dimensional systems with low or moderate entanglements.…”

Spectral and thermodynamic properties of the Holstein polaron: Hierarchical equations of motion approach

“…All these results suggest that our HEOM approach may greatly contribute to future studies of finitetemperature Holstein polaron dynamics. Using advanced propagation techniques, 98 our approach may become viable for larger or higher-dimensional systems and in multimode situations. It is interesting to note that our study is concurrent with other works aiming at extending the methods commonly applied to molecular systems (chemical-physics realm) to band-like situations (condensed-matter realm).…”

Optical response of laser-driven charge-transfer complex described by Holstein-Hubbard model coupled to heat baths: Hierarchical equations of motion approach

“…Because electron transfer is a long-range effect, an extension of the present investigation to a larger system would provide deeper insight, in particular, with regard to studies of superconductivity. Thus, to make the present approach more useful, further computational efforts need to be made to treat larger systems that consist of many sites, for example, by employing the hierarchical Schrödinger equations of motion 75 and the tensor-train method [76][77][78][79] . Such investigations are left for future work.…”